Properties

Label 2-6004-1.1-c1-0-35
Degree $2$
Conductor $6004$
Sign $-1$
Analytic cond. $47.9421$
Root an. cond. $6.92402$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.38·3-s − 2.90·5-s − 2.69·7-s + 2.69·9-s + 1.35·11-s − 3.01·13-s + 6.93·15-s + 0.160·17-s − 19-s + 6.43·21-s − 4.12·23-s + 3.45·25-s + 0.737·27-s − 4.15·29-s − 3.19·31-s − 3.23·33-s + 7.84·35-s + 10.3·37-s + 7.19·39-s + 5.30·41-s + 3.84·43-s − 7.82·45-s + 8.59·47-s + 0.280·49-s − 0.383·51-s − 8.41·53-s − 3.93·55-s + ⋯
L(s)  = 1  − 1.37·3-s − 1.30·5-s − 1.01·7-s + 0.896·9-s + 0.408·11-s − 0.836·13-s + 1.79·15-s + 0.0389·17-s − 0.229·19-s + 1.40·21-s − 0.859·23-s + 0.690·25-s + 0.142·27-s − 0.771·29-s − 0.573·31-s − 0.562·33-s + 1.32·35-s + 1.70·37-s + 1.15·39-s + 0.828·41-s + 0.585·43-s − 1.16·45-s + 1.25·47-s + 0.0401·49-s − 0.0536·51-s − 1.15·53-s − 0.530·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
Sign: $-1$
Analytic conductor: \(47.9421\)
Root analytic conductor: \(6.92402\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6004,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + T \)
79 \( 1 - T \)
good3 \( 1 + 2.38T + 3T^{2} \)
5 \( 1 + 2.90T + 5T^{2} \)
7 \( 1 + 2.69T + 7T^{2} \)
11 \( 1 - 1.35T + 11T^{2} \)
13 \( 1 + 3.01T + 13T^{2} \)
17 \( 1 - 0.160T + 17T^{2} \)
23 \( 1 + 4.12T + 23T^{2} \)
29 \( 1 + 4.15T + 29T^{2} \)
31 \( 1 + 3.19T + 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 - 5.30T + 41T^{2} \)
43 \( 1 - 3.84T + 43T^{2} \)
47 \( 1 - 8.59T + 47T^{2} \)
53 \( 1 + 8.41T + 53T^{2} \)
59 \( 1 - 6.94T + 59T^{2} \)
61 \( 1 + 4.62T + 61T^{2} \)
67 \( 1 - 9.91T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
83 \( 1 - 4.60T + 83T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 - 6.59T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.54524275633571495825493006072, −7.00077959594165574350610303270, −6.20508066901891181186176079468, −5.76545040120276570237035519582, −4.77267214743947520515310914957, −4.13788584618425938426795274195, −3.47578833399361901454444640649, −2.36083062462882708841531272475, −0.76585042365626124256548074126, 0, 0.76585042365626124256548074126, 2.36083062462882708841531272475, 3.47578833399361901454444640649, 4.13788584618425938426795274195, 4.77267214743947520515310914957, 5.76545040120276570237035519582, 6.20508066901891181186176079468, 7.00077959594165574350610303270, 7.54524275633571495825493006072

Graph of the $Z$-function along the critical line